Theorem dicvscacl 41194

Description: Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
dicvscacl.l	⊢ ≤ = (le‘𝐾)
dicvscacl.a	⊢ 𝐴 = (Atoms‘𝐾)
dicvscacl.h	⊢ 𝐻 = (LHyp‘𝐾)
dicvscacl.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicvscacl.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dicvscacl.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
dicvscacl.s	⊢ · = ( ·𝑠 ‘𝑈)

Assertion

Ref	Expression
dicvscacl	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 · 𝑌) ∈ (𝐼‘𝑄))

Proof of Theorem dicvscacl

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp3l 1201	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑋 ∈ 𝐸)
3		dicvscacl.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
4		dicvscacl.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
5		dicvscacl.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
6		dicvscacl.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
7		dicvscacl.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
9	3, 4, 5, 6, 7, 8	dicssdvh 41189	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (Base‘𝑈))
10		eqid 2736	. . . . . . . . . 10 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
11		dicvscacl.e	. . . . . . . . . 10 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
12	5, 10, 11, 7, 8	dvhvbase 41090	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘𝑈) = (((LTrn‘𝐾)‘𝑊) × 𝐸))
13	12	eqcomd 2742	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((LTrn‘𝐾)‘𝑊) × 𝐸) = (Base‘𝑈))
14	13	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (((LTrn‘𝐾)‘𝑊) × 𝐸) = (Base‘𝑈))
15	9, 14	sseqtrrd 4020	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × 𝐸))
16	15	3adant3 1132	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝐼‘𝑄) ⊆ (((LTrn‘𝐾)‘𝑊) × 𝐸))
17		simp3r 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (𝐼‘𝑄))
18	16, 17	sseldd 3983	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × 𝐸))
19		dicvscacl.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑈)
20	5, 10, 11, 7, 19	dvhvsca 41104	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (((LTrn‘𝐾)‘𝑊) × 𝐸))) → (𝑋 · 𝑌) = ⟨(𝑋‘(1st ‘𝑌)), (𝑋 ∘ (2nd ‘𝑌))⟩)
21	1, 2, 18, 20	syl12anc 836	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 · 𝑌) = ⟨(𝑋‘(1st ‘𝑌)), (𝑋 ∘ (2nd ‘𝑌))⟩)
22		fvi 6984	. . . . . 6 ⊢ (𝑋 ∈ 𝐸 → ( I ‘𝑋) = 𝑋)
23	2, 22	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → ( I ‘𝑋) = 𝑋)
24	23	coeq1d 5871	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (( I ‘𝑋) ∘ (2nd ‘𝑌)) = (𝑋 ∘ (2nd ‘𝑌)))
25	24	opeq2d 4879	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → ⟨(𝑋‘(1st ‘𝑌)), (( I ‘𝑋) ∘ (2nd ‘𝑌))⟩ = ⟨(𝑋‘(1st ‘𝑌)), (𝑋 ∘ (2nd ‘𝑌))⟩)
26	21, 25	eqtr4d 2779	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 · 𝑌) = ⟨(𝑋‘(1st ‘𝑌)), (( I ‘𝑋) ∘ (2nd ‘𝑌))⟩)
27		eqid 2736	. . . . . . . 8 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
28	3, 4, 5, 27, 10, 6	dicelval1sta 41190	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
29	28	3adant3l 1180	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
30	29	fveq2d 6909	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋‘(1st ‘𝑌)) = (𝑋‘((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
31	3, 4, 5, 11, 6	dicelval2nd 41192	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (2nd ‘𝑌) ∈ 𝐸)
32	31	3adant3l 1180	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2nd ‘𝑌) ∈ 𝐸)
33	5, 10, 11	tendof 40766	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (2nd ‘𝑌) ∈ 𝐸) → (2nd ‘𝑌):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
34	1, 32, 33	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (2nd ‘𝑌):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
35		eqid 2736	. . . . . . . . 9 ⊢ (oc‘𝐾) = (oc‘𝐾)
36	3, 35, 4, 5	lhpocnel 40021	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
37	36	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊))
38		simp2 1137	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
39		eqid 2736	. . . . . . . 8 ⊢ (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) = (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)
40	3, 4, 5, 10, 39	ltrniotacl 40582	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊) ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
41	1, 37, 38, 40	syl3anc 1372	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
42		fvco3 7007	. . . . . 6 ⊢ (((2nd ‘𝑌):((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ (℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑋 ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (𝑋‘((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
43	34, 41, 42	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((𝑋 ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = (𝑋‘((2nd ‘𝑌)‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄))))
44	30, 43	eqtr4d 2779	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋‘(1st ‘𝑌)) = ((𝑋 ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
45	24	fveq1d 6907	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → ((( I ‘𝑋) ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) = ((𝑋 ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
46	44, 45	eqtr4d 2779	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋‘(1st ‘𝑌)) = ((( I ‘𝑋) ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)))
47	5, 11	tendococl 40775	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐸 ∧ (2nd ‘𝑌) ∈ 𝐸) → (𝑋 ∘ (2nd ‘𝑌)) ∈ 𝐸)
48	1, 2, 32, 47	syl3anc 1372	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 ∘ (2nd ‘𝑌)) ∈ 𝐸)
49	24, 48	eqeltrd 2840	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (( I ‘𝑋) ∘ (2nd ‘𝑌)) ∈ 𝐸)
50		fvex 6918	. . . . 5 ⊢ (𝑋‘(1st ‘𝑌)) ∈ V
51		fvex 6918	. . . . . 6 ⊢ ( I ‘𝑋) ∈ V
52		fvex 6918	. . . . . 6 ⊢ (2nd ‘𝑌) ∈ V
53	51, 52	coex 7953	. . . . 5 ⊢ (( I ‘𝑋) ∘ (2nd ‘𝑌)) ∈ V
54	3, 4, 5, 27, 10, 11, 6, 50, 53	dicopelval 41180	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨(𝑋‘(1st ‘𝑌)), (( I ‘𝑋) ∘ (2nd ‘𝑌))⟩ ∈ (𝐼‘𝑄) ↔ ((𝑋‘(1st ‘𝑌)) = ((( I ‘𝑋) ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ (( I ‘𝑋) ∘ (2nd ‘𝑌)) ∈ 𝐸)))
55	54	3adant3 1132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (⟨(𝑋‘(1st ‘𝑌)), (( I ‘𝑋) ∘ (2nd ‘𝑌))⟩ ∈ (𝐼‘𝑄) ↔ ((𝑋‘(1st ‘𝑌)) = ((( I ‘𝑋) ∘ (2nd ‘𝑌))‘(℩𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔‘((oc‘𝐾)‘𝑊)) = 𝑄)) ∧ (( I ‘𝑋) ∘ (2nd ‘𝑌)) ∈ 𝐸)))
56	46, 49, 55	mpbir2and 713	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → ⟨(𝑋‘(1st ‘𝑌)), (( I ‘𝑋) ∘ (2nd ‘𝑌))⟩ ∈ (𝐼‘𝑄))
57	26, 56	eqeltrd 2840	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑋 ∈ 𝐸 ∧ 𝑌 ∈ (𝐼‘𝑄))) → (𝑋 · 𝑌) ∈ (𝐼‘𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950  ⟨cop 4631   class class class wbr 5142   I cid 5576   × cxp 5682   ∘ ccom 5688  ⟶wf 6556  ‘cfv 6560  ℩crio 7388  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  Basecbs 17248   ·_𝑠 cvsca 17302  lecple 17305  occoc 17306  Atomscatm 39265  HLchlt 39352  LHypclh 39987  LTrncltrn 40104  TEndoctendo 40755  DVecHcdvh 41081  DIsoCcdic 41175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-riotaBAD 38955

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-undef 8299  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-sca 17314  df-vsca 17315  df-proset 18341  df-poset 18360  df-plt 18376  df-lub 18392  df-glb 18393  df-join 18394  df-meet 18395  df-p0 18471  df-p1 18472  df-lat 18478  df-clat 18545  df-oposet 39178  df-ol 39180  df-oml 39181  df-covers 39268  df-ats 39269  df-atl 39300  df-cvlat 39324  df-hlat 39353  df-llines 39501  df-lplanes 39502  df-lvols 39503  df-lines 39504  df-psubsp 39506  df-pmap 39507  df-padd 39799  df-lhyp 39991  df-laut 39992  df-ldil 40107  df-ltrn 40108  df-trl 40162  df-tendo 40758  df-dvech 41082  df-dic 41176

This theorem is referenced by:  diclss  41196